#226204

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,7,9,6,3,5,11)(4,8,13,12,14,10,15,16)(18,23)(19,28)(21,29)(25,30)(26,27)(31,32);(1,3,6,2)(4,12,14,16)(5,9,7,11)(8,13,10,15)(17,18,22,27)(19,24,30,20)(21,26,29,23)(25,32,28,31);(2,3)(8,10)(9,11)(12,16)(17,19,29,25,22,30,21,28)(18,24,23,32,27,20,26,31);(1,4)(2,8)(3,10)(5,15)(6,14)(7,13)(9,12)(11,16)(17,20)(18,25)(19,26)(21,32)(22,24)(23,30)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133400);